In such a clamp element (cf. the prior German patent 4 018 284, corresponding to U.S. application Ser. No. 07/706,797, filed May 29, 1991 the tensioning rod is passed centrally through both the first tightening member and the second tightening member and is connected to this by a transverse pin. The tightening pins are supported radially outside the tensioning rod on the clamp members. By rotating the second tightening member relative to the first tightening member by means of the actuating and tensioning rod, the tightening pins are righted from their inclined position relative to the feed sleeve axis, i.e. they are brought into a clamping position parallel or approximately parallel to the feed sleeve axis. Since the clamping force is greatly increased by this, a correspondingly large torque is needed to turn the actuating and tensioning rod about its axis. With smaller machine vices or rapid clamping devices, for which clamp elements of the type defined are mainly used, the torque needed on tightening to turn the actuating and tensioning rod lies in a range which permits the use of a manual lever to turn the actuating and tensioning rod. For larger machine vices and clamp devices with a clamping force of 40 kN however, the torque needed to turn the actuating and tensioning rod is so large that manual force no longer suffices or the actuating lever becomes so long that it collides with parts in the vicinity of the clamp device. It should be noted that, in the clamp element of the type defined, the rotation of the actuating and tensioning rod needed for both the feed stroke and the tightening stroke should be as much less than 180.degree. as possible.